论文标题
加权马尔可夫 - 杜宾问题
The Weighted Markov-Dubins Problem
论文作者
论文摘要
在本文中，考虑了经典的马尔可夫·杜宾（Markov-Dubins）问题的变化，该问题涉及具有规定的初始和最终配置的曲率约束最低成本的路径，对于Sinistral和Dextral曲率的不同范围，以及罚款$μ_l$和$μ_r$和$μ_r$，分别用于Sinistral and Dextral toxtral和Dextral to。解决的问题概括了经典的马尔可夫 - 杜宾问题和不对称的罪行/右旋马尔可夫 - 杜宾问题。拟议的配方可用于建模无人机（UAV），由于高度损失而造成的罚款或由于硬件故障或环境条件而导致的罪行和右转弯的无人机，因此与转弯相关的罚款。使用最佳控制理论，本文的主要结果表明，最佳路径最多属于$ 21 $的候选路径，每个路径最多包括五个部分。与经典的马尔可夫·杜宾（Markov-Dubins）问题不同，$ ccc $路径是经典马尔可夫·杜宾（Markov-Dubins）问题的候选路径，对于加权的马尔可夫·杜宾（Markov-Dubins）而言并不是最佳的。此外，在$μ_l$和$μ_r$接近零时，获得的加权马尔可夫 - 杜宾（Markov-Dubins）问题获得的候选路径列表将减少到标准$ CSC $和$ CCC $路径以及相应的退化路径。
In this article, a variation of the classical Markov-Dubins problem is considered, which deals with curvature-constrained least-cost paths in a plane with prescribed initial and final configurations, different bounds for the sinistral and dextral curvatures, and penalties $μ_L$ and $μ_R$ for the sinistral and dextral turns, respectively. The addressed problem generalizes the classical Markov-Dubins problem and the asymmetric sinistral/dextral Markov-Dubins problem. The proposed formulation can be used to model an Unmanned Aerial Vehicle (UAV) with a penalty associated with a turn due to a loss in altitude while turning or a UAV with different costs for the sinistral and dextral turns due to hardware failures or environmental conditions. Using optimal control theory, the main result of this paper shows that the optimal path belongs to a set of at most $21$ candidate paths, each comprising of at most five segments. Unlike in the classical Markov-Dubins problem, the $CCC$ path, which is a candidate path for the classical Markov-Dubins problem, is not optimal for the weighted Markov-Dubins problem. Moreover, the obtained list of candidate paths for the weighted Markov-Dubins problem reduces to the standard $CSC$ and $CCC$ paths and the corresponding degenerate paths when $μ_L$ and $μ_R$ approach zero.
